![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mircgrextend | Structured version Visualization version GIF version |
Description: Link congruence over a pair of mirror points. cf tgcgrextend 28508. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirtrcgr.e | ⊢ ∼ = (cgrG‘𝐺) |
mirtrcgr.m | ⊢ 𝑀 = (𝑆‘𝐵) |
mirtrcgr.n | ⊢ 𝑁 = (𝑆‘𝑌) |
mirtrcgr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirtrcgr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
mirtrcgr.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
mirtrcgr.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mircgrextend.1 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) |
Ref | Expression |
---|---|
mircgrextend | ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . 2 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | mirtrcgr.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | mirtrcgr.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirtrcgr.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐵) | |
10 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircl 28684 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | mirtrcgr.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
12 | mirtrcgr.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
13 | mirtrcgr.n | . . 3 ⊢ 𝑁 = (𝑆‘𝑌) | |
14 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircl 28684 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃) |
15 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mirbtwn 28681 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴)) |
16 | 1, 2, 3, 4, 10, 6, 5, 15 | tgbtwncom 28511 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴))) |
17 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mirbtwn 28681 | . . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋)) |
18 | 1, 2, 3, 4, 14, 12, 11, 17 | tgbtwncom 28511 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋))) |
19 | mircgrextend.1 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌)) | |
20 | 1, 2, 3, 4, 5, 6, 11, 12, 19 | tgcgrcomlr 28503 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋)) |
21 | 1, 2, 3, 7, 8, 4, 6, 9, 5 | mircgr 28680 | . . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴)) |
22 | 1, 2, 3, 7, 8, 4, 12, 13, 11 | mircgr 28680 | . . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋)) |
23 | 20, 21, 22 | 3eqtr4d 2785 | . 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋))) |
24 | 1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23 | tgcgrextend 28508 | 1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 cgrGccgrg 28533 pInvGcmir 28675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 df-mir 28676 |
This theorem is referenced by: mirtrcgr 28706 |
Copyright terms: Public domain | W3C validator |